Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...

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121 views

Inverse of $f(x)=3^x+2^x$

I'm tring to find inverse of $f(x)=3^x+2^x$ but I don't have any clue. I tried to $$y=2^x((3/2)^x+1)$$ $$\ln y=\ln2^x+\ln((3/2)^x+1)$$ $$\ln y= x \ln2+\ln((3/2)^x+1)$$ but I can't continue
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2answers
32 views

Given $2\arcsin(x)-3\arccos(x)=\frac{\pi}6 $, find the value of x.

I know that $\arcsin(x) + \arccos(x) = \frac{\pi}2$, but how to use that to solve the following question? $$2\arcsin(x)-3\arccos(x)=\frac{\pi}6 $$
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1answer
26 views

Does an $x \in \mathbb Z$ with $xa\equiv_nb$ exist, if $gcd(a,n) $ divides $b$?

Does an $x \in \mathbb Z$ with $xa\equiv_nb$ exist, if $gcd(a,n) $ divides $b$ ? My idea: \begin{aligned} & xa &\equiv_n& \quad b \cr \Leftrightarrow \quad& x& \equiv_n& \quad ...
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1answer
19 views

Using inverses to find solutions.

So upon solving some trigonometric equations, I found myself using the following method often:$$f[g(x)]=h(x)$$$$f[g(g^{-1}(x))]=h[g^{-1}(x)]$$$$f(x)=h[g^{-1}(x)]$$Which is how I usually find $f(x)$ ...
2
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1answer
54 views

Finding the inverse of a “bow-shaped” matrix

Consider the matrix $$A = \begin{bmatrix} n_{+} & n_1 & n_2 & n_3 & \cdots & n_{r-1} \\ n_1 & n_1 & 0 & 0 & \cdots & 0 \\ n_2 & 0 & n_2 ...
2
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3answers
57 views

Inverse trigonometric Problem

For any $x \in [-1,0) \cup (0,1]$, how can I prove that: $$\sin^{-1}(2x\sqrt{1-x^2})=2\cos^{-1}x$$ Also, can someone explain to me how to understand the graphs of $sin$ and $cos$ functions?
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1answer
30 views

Jacobian of inverse of matrix $A(x) \in \mathcal{M}_{7\times7}$?

I have a matrix $A(x)$ where $x\in \mathbf{R}^{7}$. I have to calculate $\frac{\partial}{\partial x}A(x)^{-1}$ and then I will evaluate it at some $x_{0}$. Now this matrix is very dense so its not ...
0
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0answers
39 views

Inverse Laplace transform of complicated function

I have a Laplace transformed function that I'd like to transform back. It's quite a complex function however, which is why I am stuck: $$C(x,s) = ...
1
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0answers
32 views

Derivative of inverse matrix

Suppose $\Omega \left( \mathbf{\alpha }\right) $ is a $T\times T$ full rank matrix where $\mathbf{\alpha }$ is a $p\times 1$ vector, then what's the exact expression for $\frac{\partial \Omega ...
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6answers
7k views

Finding the inverse of $h(x) = 3^x$

most of the time I know how to find the inverse of a function (make it equal $y$, solve for $x$ and then swap $x$ and $y$), but I have no idea how to do that for this one, so any help would be great: ...
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2answers
78 views

Prove that $\ln$ and $\exp$ are inverses

If we take the definitions of $\exp$ and $\ln$ as follows: $\exp(x) = {\large\sum\limits_{i=0}^\infty} \dfrac{x^i}{i!}$ $\ln(x) = {\large\int_1^x} \dfrac1t\ dt$ how could we prove that these ...
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4answers
11k views

Why are nonsquare matrices not invertible?

I have a theoretical question. Why are non-square matrices not invertible? I am running into a lot of doubts like this in my introductory study of linear algebra.
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2answers
34 views

Proof about Diagonalization of A

The question asks WHY is it true that $$A^{n} = PD^{n}P^{-1}$$ I can never do proper proving in algebra; what I almost know for sure is that a proof by induction is the way to go here. But how do you ...
0
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1answer
56 views

$2 \ne 3$, but where's my error?

In $\mathbb{Z}_6$, $3^3 = 3^{-3}$ since $3^{-3} = 3^{6-3} = 3^3$. Thus $(3)^3 = (3^{-1})^3=2^3=2$. But also $3^3 = 3$ in $\mathbb{Z}_6$. Where's my error? Sorry for this question, but I think I got ...
1
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1answer
22 views

Inverse function on given sets

My question is: Given sets $A$ = {$a_1, a_2$} and B = {$b_1$}, let the function $f$ from $A$ to $B$ be given by the following set of ordered pairs, $f$ = { ($a_1, b_1$), ($a_2, b_1$) }. If $f$ has an ...
2
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1answer
79 views
0
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1answer
43 views

If the function $f(x)=ax+b$ has its own inverse,then the ordered pair $(a,b)$ can be

If the function $f(x)=ax+b$ has its own inverse,then the ordered pair $(a,b)$ can be $(A)(1,0)\hspace{1cm}(B)(-1,0)\hspace{1cm}(C)(-1,1)\hspace{1cm}(D)(1,1)$ This is a more than one options correct ...
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0answers
12 views

Prove that $\frac{1}{[Z^{-1}]_{kk}}=\frac{\text{det}Z} {\text{det}Z_{kk}}=\text{det}Z_{kk}^{\text{SC}}$, $Z_{kk}^{\text{SC}}$ is the Schur complement

Suppose $Z$ is a complex (Wishart) matrix. Let $a=\frac{1}{[Z^{-1}]_{kk}}$, where $Z^{-1}$ is the inverse of $Z$ and $[Z^{-1}]_{kk}$ represents the $(k,k)$-th entry of $Z^{-1}$. When I was reading ...
0
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1answer
28 views

Matrix Inversion distribution

How do you distribute the inversion in $(A^TA+\lambda I)^{-1}A^Ty$ assuming $A$ is a $n \times n$ square invertible matrix, $y$ is a vector with the dimension of $n$, and $\lambda$ is a constant?
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2answers
51 views

Finding the value of Inverse Trigonometric functions beyond their Real Domain

I wanted to ask how can we calculate the values of the inverse of trigonometric functions beyond their domain of definition, for example $\arcsin{2}$ beyond its domain of ...
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1answer
35 views

Can we always for an invertible matrix $M$ find real number $\alpha \neq 0$ such that $M+\alpha$ is invertible?

I do not know enough about matrices, maybe only enough to be able to create question like this one, but I would like to see an answer. Let $a_{ij}$ be some element of invertible $n\times n$ matrix ...
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0answers
15 views

Implicit function theorem, what is the meaning of invertible linear operator?

I have to show that around $(1,-1,0)$(have to find the neighborhood as well), $x,y$ are determined uniquely by $z$ given $x+yz-z^3=1, x^3-xz+y^3=0$. What I did so far is: $f:\Bbb{R}^2\times\Bbb{R}\to ...
2
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1answer
26 views

b is the inverse of a $( \mod 11)$

Let a and b be numbers in the set $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8 , 9, 10\}$ such that b is the inverse of a $(\mod11)$ and a and b are not equal. How many such subsets $ \{a, b\}$ of S are there?
2
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1answer
61 views

How can I determine B-inverse from an optimal tableau of a LP?

(This is NOT a homework question, I am reviewing for my upcoming exam) Given this linear program: and this optimal tableau: I am attempting to determine $B$ inverse using the table above. From ...
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3answers
98 views

How to Prove that this Function is not $1-1$

I'm trying to show that the function $$f(x)= \dfrac{x}{4}+x^2\sin\left(\dfrac{1}{x}\right)$$ is not $1-1$ for any neighborhood of $0$. I know that what I have to do is find two different points that ...
0
votes
1answer
13 views

Production Model x=Cx+d — Use Inverse Matrix

Question: Consider the production model x = Cx + d for an economy with two sectors, where C= 0.0 0.5 0.6 0.2 and d= 50 30 Use an inverse matrix to determine the production level ...
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74 views

Inverse of generalized arrow matrix $A = M^T * M + I$

If we have the following linear system: Ax=b And matrix A is created by multiplying a rectangular matrix with it's transpose: $A = M^T * M + I$ What is the best method to solve for x for different b ...
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1answer
31 views

Inverse Laplace Transform of $\frac{1}{\sqrt{s+a}+\sqrt{s+b}}$

I need to calculate the inverse laplace of: $$F(s)=[\frac{1}{\sqrt{s+a}+\sqrt{s+b}}] \qquad \qquad (s>-a\quad ;\quad s>-b;\quad a\neq b) $$
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2answers
31 views

Inverse of an exponential function

I am having difficulties forming the inverse of this $f(x) = 3 \cdot2^{3x+1} \cdot 5^{3x-1}$. What I have done so far: $3 \cdot 2^{3y} \cdot 2^1 \cdot 5^{3y}\cdot5^{-1} \Leftrightarrow 3\cdot 2\cdot ...
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3answers
47 views

Is $f(x) = e^x$ one-to-one if $f:\mathbb{R} \rightarrow \mathbb{R}$?

My book says that $f(x) = e^x$ is not invertible from the set of real numbers to the set of real numbers. But I disagree since $f(x) = e^x$ is injective with this given domain and codomain and ...
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0answers
23 views

Inverting a sparse Matrix

I have a sparse, square, symmetric matrix with the following structure: (Let's say the size of the matrix is N x N) the structure of the sparse matrix Here, the area under the blue stripes is the ...
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0answers
16 views

Proof of the inverse function theorem in van der Vaart

I have a question regarding the proog of lemma 4.3 in van der Vaart at p.36 ...
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0answers
29 views

Proving inverse function theorem

I'm asking for an help to understand the proof of the inverse function theorem, in particular one part of it represented by Lemma 4.2 in van der vaart p.36 (you can find it here ...
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1answer
22 views

Inverting this function

I have to invert this function: $$f: \mathbb{N} \rightarrow \mathbb{N}, f(n) = \begin{cases} n+1, & \text{if $n$ is odd} \\ n-1, & \text{if $n$ is even} \end{cases}$$ But I am not able to ...
3
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1answer
49 views

Show that $P$ is symmetric.

Let $P$ = $A(A^TA)^{-1}A^T$, where A is an m x n matrix with rank $n$. I feel like this is wrong, but here is my attempt: $A(A^TA)^{-1}A^T$ = $AA^{-1}(A^T)^{-1}A^T$ = $I$ And $I^T$ = $I$, so the ...
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1answer
42 views

Show that $P^2$ = $P$

Let $P$ = $A(A^TA)^{-1}A^T$, where A is $m \times n $ 0f rank $n$. This is the projection matrix, right? Every site I've been on says that this is the projection matrix such that $P^2$ = $P$, but ...
0
votes
1answer
48 views

Inverse Laplace Transform and error function

Express your answer in terms of the error function: $$L^{-1}\left[\frac{1}{\sqrt{s^3+as^2}}\right]$$ Clue: $\qquad L\left[\frac{1}{\sqrt{t}}\right]=\sqrt\frac{π}{s} \qquad , \qquad s>0$ Error ...
3
votes
2answers
7k views

Finding the inverse of a matrix by elementary transformations.

While using the elementary transformation method to find the inverse of a matrix, our goal is to convert the given matrix into an identity matrix. We can use three transformations:- 1) Multiplying ...
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37 views

Getting the derivative of the inverse of a function

Given $f(x)$, how would I find $(f^{-1})'(x)$? As an example how would I find that for this problem: $f(x) = 4x^3 + 5x + 2$
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1answer
22 views

Limits, Determinants and Inversion of a matrix-valued function

Suppose I have a matrix-valued, continuous function $$A\colon [0,\infty) \to \mathbb R^{n\times n},\qquad h\mapsto A(h).$$ I know that for the limit $h\to 0$ the matrix is invertible: ...
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2answers
52 views

a practical question about matrix derivative with inverse and chain rule: dimension mismatch

Recently, I was trying to take the following derivative $$ \dfrac{\partial (X^TV^{-1}X)^{-1}}{\partial V} $$ I was referring to matrix cookbook to solve it, where I found several useful equations: ...
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1answer
38 views
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3answers
106 views

Why does Arccos(Sin(x)) look like this??

I can kind of understand the main direction (slope) of $y$ over the different $x$ intervals, but I can't figure out why the values of $y$ take on the shape of straight lines and not curves looking ...
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0answers
27 views

Prove a function with a positive definite Jacobian is 1-1

Suppose that $f:\mathcal{O}\subset\Bbb{R}^n\to\Bbb{R}^n, \mathcal{O}$ open and convex, $f\in C^1(\mathcal{O})$, and the symmetric part of the Jacobian matrix, $\frac{(Df)(x)+(Df)^T(x)}{2}$, is ...
0
votes
3answers
25 views

Consider $f : \mathbb{N} → \mathbb{Z}$ defined as $f (n) = \frac{(−1)^n (2n−1)+1}{4}$. Find its inverse.

I cannot find an inverse of this function for f(n) = x, where x is an integer, that gives out a natural number. Some guidance would be very helpful... I already know the function is bijective so there ...
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2answers
97 views

There exists a map $f:\mathbb{N}\to\mathbb{N}$ that is injective, but not surjective.

Prove: There exists a map $f:\mathbb{N}\to\mathbb{N}$ that is injective, but not surjective. I immediately thought of the function $f(x)=2x$ It is trivial to prove the "not surjective part" ...
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1answer
29 views

Properties of adjoint matrix in a finite dimensional inner product space

let $V$ be a finite dimensional inner product space. Let $T$ be a linear operator on $V$. Prove that there exists an invertible linear operator $U$ such that $U^{-1}TT^*U = T^*T $ where $T^*$ is ...
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0answers
21 views

Hessian for inverse probit link

I'm trying to calculate Hessian and Fisher Information for binomial model using inverse probit link, Suppose likelihood function is $L(\pi)=\prod\limits_{i=1}^n \pi_i^{y_i}(1-\pi_i)^{1-y_i}$ and ...
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3answers
33 views

For which values of is the following matrix invertible and what is its inverse?

For which values of is the following matrix invertible and what is its inverse? $$A =\begin{bmatrix}1 & 1 &0\\1 & 2 & 2\\1 & 2 & \lambda\end{bmatrix}$$ If someone can please ...
0
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4answers
39 views

Inverse of function - Difficulty solving [closed]

$$y=\sqrt{35\tan (\frac{\pi }{180}x)}$$ I have a really hard time finding the inverse of this particular function. Can anyone shine some light on why that may be, or alternatively solve it if I'm ...